COMPENSATED COMPACTNESS, PARACOMMUTATORS, AND HARDY-SPACES

Let B-1:R-n x R-N1 --> R-m1, B-2:R-n x R-N2 --> R-m2 and Q:R-m2 --> R- m1 be bilinear forms which are related as follows: if mu and nu satisf y B-1(xi, mu) = 0 and B-2(xi, nu) = 0 for some xi not equal 0, then mu (tau)Q nu = 0. Suppose p(-1) + q(-1) = 1. Coifman, Lions, Meyer and Se mmes proved that, if u is an element of L-p(R-n) and v is an element o f L-q(R-n), and the first order systems B-1(D, u) = 0, B-2(D, v) = 0 h old, then u(tau)Qv belongs to the Hardy space H-1(R-n), provided that both (i) p = q = 2, and (ii) the ranks of the linear maps B-j(xi,.):R- Nj --> R-m1 are constant. We apply the theory of paracommutators to sh ow that this result remains valid when only one of the hypotheses (i), (ii) is postulated. The removal of the constant-rank condition when p = q = 2 involves the use of a deep result of Lojasiewicz from singula rity theory. (C) 1997 Academic Press.